June 26, 2011

Hopf algebra structure on 3-manifolds!?

The main question in quantum topology has always been a curious one- “What do quantum invariants mean topologically?”
For 3-manifolds, part of the mystery has been that quantum invariants, such as the Reshetikhin-Turaev and Turaev-Viro invariants, are constructed using the representation theory of Hopf algebras. But wait a minute- 3-manifolds don’t admit a natural Hopf algebra structure, do they? What’s going on? Why should Hopf algebras have anything at all to say about 3-manifold topology? Why should quantum invariants exist?
As is often the case in mathematics, the first step in understanding is to identify the right category in which to ask the question. Quantum invariants give information about closed 3-manifolds sure enough, but their main property is locality, meaning that they take fixed values on small simple pieces, which we later glue together. Like a lego tower, we build our 3-manifold by piecing together the blocks. So the right category to be considering really isn’t the category of completed lego towers, or closed 3-manifolds, but rather a category of lego blocks, which are later to be pieced together.
There are many categories of lego blocks we might choose, depending on how we might want to slice up our 3-manifold. But, because the invariants themselves are based on Hopf algebras, maybe the guiding principle should be that our category of lego blocks should naturally admit a (braided) Hopf algebra structure.
In the 1990’s, Crane and Yetter [1], and independently Kerler [2] noticed that the category of connected surfaces with one boundary component, and their cobordisms, might be just what we’re looking for. It admits a Hopf algebra structure, and its blocks are something you can chop closed 3-manifolds up into. But identifying that structure precisely is difficult.
In [3], Kerler constructed a category with generators and relations, and a surjective functor from to . The category is all you could ever desire- it represents a braided Hopf algebra of the right sort, so conceptually it would be a lovely natural domain for a quantum invariant. Except that it’s not isomorphic to because it doesn’t have enough relations. Kerler posed the problem of identifying all the relations which must be imposed on to make it isomorphic to .
What a delightfully deep problem! Let’s pause a moment to take in what it says. We would like to find a purely algebraic category, given in terms of generators and relations, which is to recapture all of 3-manifold topology. And if you find it, quantum invariants of 3-manifolds suddenly make sense! Thinking about things this way, the problem is of the same species as the Poincaré Conjecture and Geometrization. So one might expect this problem to also be very difficult.
Well, Bobtcheva and Piergallini [4] found additional relations, but the point of this post is to publicise Asaeda’s announcement that Kazuo Habiro might have found all of them! What an incredible breakthrough! This announcement, together with the list of all 26 relations, is to be found in a recent paper by Marta Asaeda [5], who builds what is to be a Turaev-Viro type TQFT directly from (with the extra relations).
This is something to be excited about- well, I’m excited about it. 26 relations is a lot of relations, but a new fully algebraic picture of 3-manifolds (which is what we’ll be getting) is not bad at all if you ask me. Quantum topology could soon be ready to hit the big time.

If you want to build up a 3-manifold by gluing together cobordisms between punctured surfaces, what kind of restrictions are there on how you are allowed to glue the cobordisms together? The cobordism of the surface restricts to a cobordism of the surfaces’ boundaries – Do these need to be glued to each other?

Nice question! The horizontal gluing is topologically a bit surprising, because you’re gluing along only part of the boundary. The reference for the concrete setup I’m about to describe is Habiro’s Bottom Tangles and Universal Invariants. First, you set up the surface in explicit coordinates in as with handles attached uniformly in the x direction. A cobordism between these is a cube with handles added on top (the top surface) and drilled out of the bottom (the bottom surface).
Now you can set up as a monoidal category in the strict sense. Vertical composition is stacking, i.e. “thread the tubes through the holes”, and horizontal composition is side-by-side juxtaposition, i.e. “place the blocks side by side”. This is exactly what you expect from a Hopf algebra, or from a set of lego. To get a closed surface, you cap-off by attaching to the boundary; which kind of sucks, but that’s how quantum topology tends to work- you have to close up objects “by hand”.
This is topology imitating algebra; I wonder how one can bring this picture closer to the kinds of things we consider classically. That’s a challenge even for the much simpler setup of knot and tangle diagrams…